1. Field of the Invention
This invention relates to visualization and graphical analysis of functions considered on multi-dimensional domain especially for multi-objective research objects.
2. Description of the Prior Art
In the 17th century R. Descartes discovered that it was very useful to combine two variables in a horizontal/vertical diagram (a Cartesian diagram/coordinate system). Moreover, this discovery made it much easier to see patterns than it had been previously. Later on the coordinate system was widely utilized for data and functions"" visualization.
In recent years many different methods have been developed for multi-dimensional data analysis and visualization. But there are no new methods of visualization for multi-dimensional functions.
Consider a function of N independent variables y=f(x1, x2, . . . , xN). All known methods of graphical analysis for such functions are based on a very old tradition (at least 300 years old). Such methods reduce the number of independent variables by assigning a number of constant values to some of the variables. The resulting function is a function of one or two variables, and can be drawn without any difficulties. Consider the two most popular methods of graphical visualization for the following functions:
1. Assign constant values to all independent variables except of the first one: x2=p1;. x3=C3 . . . , xN=CN, where 1 is a parameter value, and than draw a one-dimensional dependency y=f1(x1) in axes (y,x1); assign another parameter value x2=p2 and draw another one-dimensional dependency y=f2(x1) in axis (y,x1), and so on; see FIG. 1A;
For example, for function
y=sin(x1)+cos(x2)+log(x3)
assigned variables are x2=0.2; x3=1; so y=sin(x1)+a, where a=cos(0.2)+log(1)=0.980; As a result one-dimensional function y=0.980+sin (x1) is drawn. After that parameter x2=0.3, a=cos(0.3)+log(1) =0.955, y=0.955+sin (x1) is drawn, and so on; see FIG. 1A;
2. Assign constant values to all independent variables except for two: xk and xl, and draw a contour in axes (xk, xl) for fkl(xk, xl)=Y1, than for fkl(xk, xl)=Y2, and so on. As result a contour for every level Y1, Y2, . . . ; is created. See FIG. 1B. For example, for the same function (1) all points (x1,x2) needed to be drawn belong to the solution of following equation:
sin(x1)+cos(c2)+b=Y1,
where b=log(c3);
Solving any additional problem for this approach brings up a necessity to use special analytical or numerical methods to determine all points (xk, xl) that belong to the equation.
The major difficulty with the above described method of graphical visualization is that it does not permit one to visualize the function""s behavior in the entire multidimensional domain of function determination. Usually only the two-dimensional cuts of the function can be seen, and the graphical image depends on the constant values being used to determine the cuts. Typically, there is no simple systematic way (except complex mathematical or numerical analysis) to determine the constants"" values allowed to see graphically the most interesting effects of the function""s behavior. Therefore, the most important information is often lost and the complete picture can never be seen.
In other words the prior art permits one to graphically analyze functions with just one or two arguments. If a function depended on three or more arguments, the number of independent variables would need to be reduced. That is the reason why it is necessary to assign a constant value for all variables except for two. This is a strong constraint of graphical visualization.
In an actual situation a researcher may need to analyze graphically several multidimensional functions at the same time. The amount of dimension summarization of such a visualization task is much higher than for just one function. Therefore, the existing methods become even less effective.
A method of visualization and graphical analysis of functions considered in a multi-dimensional domain is provided. The principal approach is to approximate a function (or plurality of functions) by a dataset created by values of the function(s) in the points of a uniformly distributed sequence, and analyze the dataset graphically by dividing the dataset in two or more non-intersecting subsets via split-criteria and drawing each subset in a distinct color on plurality of two- and three-dimensional projections.
In accordance with the present invention, a data analysis system and method are provided that substantially eliminate or reduce disadvantages of previously developed data analysis tools.
Accordingly, this invention provides an interactive method of graphical analysis for a function (or a system of functions) considered in a multidimensional domain. The method is based on the approximation of the function by its values on the points of a uniformly distributed sequence and further graphical analysis of the dataset. In this way the function can be analyzed on the entire multidimensional domain without the necessity to artificially reduce dimension of the domain by assigning constant values to some of the function""s parameters. In other words, instead of using two- or three-dimensional cuts (with no information between them) we can use two- or three-dimensional projections of n-dimensional domain without losing any information. Usage of several charts in the interactive mode is sufficient, because it allows one to find a combination of axes for every particular chart that visualize the functions"" behavior as a pattern.
Another object of the invention is to provide new methods of graphical analysis by introducing split-criteria dividing the approximating dataset into two or more subsets and to color points of each subset differently, thus making a specific function""s behavior visible graphically. It is possible to create an unlimited number of different split-criteria, and each one will generate a new method of graphical analysis.
Still further objects and advantages will become apparent from a consideration of the ensuing description and accompanying drawings.